This is in contrast to an observable which combines the values of fields at different spacetime points other than by forming spacetime averages. For instance if x≠y∈Σx \neq y \in \Sigma are two distinct spacetime points, then

Φ↦Φ(x)Φ(y)
\Phi \mapsto \Phi(x) \Phi(y)

is a smooth function on the space of fields, hence an observable, but it is not a local observables.